Ancient solutions to free boundary mean curvature flow

TL;DR

This paper characterizes ancient solutions to free boundary mean curvature flow, revealing their structure near minimal hypersurfaces and constructing foliations to understand their geometric behavior.

Contribution

It establishes rigidity results, classifies ancient solutions near minimal hypersurfaces, and constructs foliations for detailed geometric analysis.

Findings

Ancient solutions form I-parameter families from minimal hypersurfaces.

Backward convergence to minimal hypersurfaces is exponential for certain solutions.

Constructed free boundary mean convex foliations around unstable minimal hypersurfaces.

Abstract

We establish rigidity results for ancient solutions to the free boundary mean curvature flow in manifolds with convex boundary. In particular, we show that any free boundary minimal hypersurface of Morse index I admits an I-parameter family of ancient solutions that emanate from it. Moreover, among ancient solutions that backward converge exponentially fast to the minimal hypersurface, these exhaust all possibilities. Additionally, we construct a smooth free boundary mean convex foliation around an unstable free boundary minimal hypersurface that enables us to provide a more detailed geometric description of mean-convex ancient solutions that backward converge to that minimal surface.